Block Renormalization Group Approach for Correlation Functions of Interacting Fermions

Inspired by a decomposition of the lattice Laplacian operator into massive terms (coming from the use of the block renormalization group transformation for bosonic systems), we establish a telescopic decomposition of the Dirac operator into massive terms, with a property named orthogonality between scales. Making a change of Grassmann variables and writing the initial fields in terms of the eigenfunctions of the operators related to this decomposition, we propose a multiscale structure for the generating function of interacting fermions. Due to the orthogonality property we obtain simple formulas, establishing a trivial link between the correlation functions and the effective potential theories. In particular, for the infrared analysis of some asymptotically free models, the two point correlation function is written as a dominant term (decaying at large distances as the free propagator) plus a correction with faster decay, and the study of both terms is straightforward once the effective potential theory is controlled.